Let R be a commutative Noetherian ring with nonzero identity, let a be an ideal of R, and let M and N be two (finitely generated) R-modules. We prove that \( H_{\mathfrak{a}}^i\left( {M,N} \right) \) is a minimax \( \mathfrak{a} \)-cofinite R-module for all i < t, t ∈ \( {{\mathbb{N}}_0} \), if and only if \( H_{\mathfrak{a}}^i{{\left( {M,N} \right)}_p} \) is a minimax \( {R_{\mathfrak{p}}} \) -module for all i < t. We also show that, under certain conditions, \( \mathrm{Ho}{{\mathrm{m}}_R}\left( {\frac{R}{\mathfrak{a}},H_{\mathfrak{a}}^t\left( {M,N} \right)} \right) \) is minimax (t ∈ \( {{\mathbb{N}}_0} \)). Finally, we study necessary conditions for \( H_{\mathfrak{a}}^i\left( {M,N} \right) \) to be \( \mathfrak{a} \)-minimax.
Similar content being viewed by others
References
M. Aghapournahr and L. Melkersson, “Finiteness properties of minimax and coatomic local cohomology modules,” Arch. Math., 94, 519–528 (2010).
J. Azam, R. Naghipour, and B. Vakili, “Finiteness properties of local cohomology modules for a-minimax modules,” Proc. Amer. Math. Soc., 137, 439–448 (2009).
R. Belshoff, E. E. Enochs, and J. R. G. Rozas, “Generalized Matlis duality,” Proc. Amer. Math. Soc., 128, 1307–1312 (2000).
M. H. Bijan-Zadeh, “A common generalization of local cohomology theories,” Glasgow Math. J., 21, 173–181 (1980).
N. Bourbaki, Elements of Mathematics Commutative Algebra, Springer-Verlag (1989).
M. P. Brodmann and R. Y. Sharp, Local Cohomology. An Algebraic Introduction with Geometric Applications, Cambridge University Press (1998).
K. Divaani-Azar and M. A. Esmakhani, “Artinianess of local cohomology modules of ZD-modules,” Comm. Algebra, 33, 2857–2863 (2005).
J. Herzog, “Komplex Auflösungen und Dualität in der lokalen Algebra,” Habilitationsschrift, University Regensburg (1970).
K. B. Lorestani, P. Schandi, and S. Yassemi, “Artinian local cohomology modules,” Can. Math. Bull., 50, 598–602 (2007).
L. Melkersson, “Modules cofinite with respect to an ideal,” J. Algebra, 285, 649–668 (2005).
J. Rotman, An introduction to Homological Algebra, Academic Press (1979).
P. Rudolf, “On minimax and related modules,” Can. J. Math., 49, 154–166 (1992).
H. Zochinger, “Minimax modules,” J. Algebra, 102, 1–32 (1986).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 6, pp. 796–801, June, 2013.
Rights and permissions
About this article
Cite this article
Kianezhad, A., Taherizadeh, A.J. Finiteness Properties of Minimax and \( \mathfrak{a} \)-Minimax Generalized Local Cohomology Modules. Ukr Math J 65, 884–890 (2013). https://doi.org/10.1007/s11253-013-0825-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0825-3